Toby... there is a slight difference here between spaces and groups. But if referring to groups, I have done this more than a few times. Once in the group, go to the "manage" drop down menu and select "members". The next page should show you all members of the group and their role. From there you can both make yourself an admin of the group, while also making another user (or current owner) a normal member, should you desire.

From what I can see from your screen shot, even though you are a full access admin, groups live "outside" your domain. The person who created the group, or an administrator of the group has to grant you administrator access to the group before you can modify anything related to it.

Here is an example of a group i created, note that since i am the creator, i cannot change my own privileges